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Theorem dmdi2 30085
 Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
Assertion
Ref Expression
dmdi2 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))

Proof of Theorem dmdi2
StepHypRef Expression
1 dmdi 30083 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))
2 eqimss2 3999 . 2 (((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))
31, 2syl 17 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114   ∩ cin 3907   ⊆ wss 3908   class class class wbr 5042  (class class class)co 7140   Cℋ cch 28710   ∨ℋ chj 28714   𝑀ℋ* cdmd 28748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-iota 6293  df-fv 6342  df-ov 7143  df-dmd 30062 This theorem is referenced by: (None)
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